Plus Two Maths Chapter Wise Questions and Answers Chapter 4 Determinants are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 4 Determinants.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Chapter Wise Questions |

Chapter | Chapter 4 |

Chapter Name | Determinants |

Number of Questions Solved | 45 |

Category | Plus Two Kerala |

## Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 4 Determinants

**Short Answer Type Questions**

**(Score 3)**

Question 1.

a. If A² – A + I = 0, A^{-1} = ?

b. If A = , then find | A |

Answer.

a. A² – A + I = 0

multiplying by A^{-1}

⇒ A – I + A^{-1} = 0

∴ A^{-1} = I – A

b. |A| = 34

Question 2.

Consider

(i) Apply R3 → R1 + R3

(ii) Show that

∆ = (α – β)(β – γ)(γ – α)(α + β + γ)

Answer.

Question 3.

If

Find the value of x.

Answer.

3x – x² = -2-8

x² – 3x – 10 = 0

x = 5 or -2

Question 4.

Calculate |A|.

ii. Find |adjA|.

[Hint: Using the property A(adjA)=|A|]

iii. Find |3A|

Answer.

i. |A| = -28

ii. |Adj A| = |A|²

= (28)² = 784

iii. |3A| = 27 x |A| = 27 x -28 = -756

Question 5.

If , then show that |2A| = 4 |A|.

Answer.

Question 6.

If

, then find the value of x

Answer.

x² – 36 = 36 – 36

x² = 36

x = ±6

Question 7.

Using the property of determinants and without expanding, prove that

Answer.

Question 8.

If a,b,c are in A.P, find value of

Answer.

Question 9.

Find the value of k if the area of a triangle with vertices (k,4), (2,-6) and (5,4) is 35 square unit.

Answer.

Area of the triangle

Question 10.

If

then,

i. Find the 2×2 matrix A.

ii. Find A².

Answer.

**Long Answer Type Questions**

**(Score 4)**

Question 1.

Show that

Answer.

Question 2.

Consider the points A(-2,-3),B(3,2) and C(-1,-8)

i. Find the area of ∆ABC.

ii. Find third vertices of any other triangle with same area and base AB.

Answer.

i. Area of ∆ABC =

Area = 15 sq.units

ii. Base line is fixes as AB for the third point. Choose any value for x and only, and find the y coordinate for that x-coordinate accordingly. (or vice versa)

For example if we choose x = 1

Question 3.

If

(i) Find AdjA.

(ii)Find A^{-1}

(iii) Verify that (Adj A)^{-1} = (Adj A^{-1})

Answer.

(i) Co-factor matrix is

Question 4.

A mixture is to be made of three foods A, B, C. The three foods A, B, C contains nutrients P, Q, R as shown below:

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R?

Answer.

Converting into linear equations

x+3y+4z=8 (for P)

2x+y+2z=5 (for Q)

5x+y+z=7 (for R)

In matrix form, (AX = B)

Question 5.

Given

(i) Apply R1→R1+R2+R3.

(ii) Take (x + y + z) common from R1.

(iii) Perform C1→C1 – 2C3, C2→C2 – C3 .

(iv) Prove that ∆ = (x+y+z)(x-z)²

Answer.

Question 6.

Let

(a)Find |A|

(b) Find AdjA.

(c) Verify that A.AdjA = |A|.I

Answer.

(a) |A| = -1(3 – 6) – 2(3 – 9) + 4(2 – 3)

= -1 x – 3 – 2 x – 6 + 4 x – 1

= 3 + 12 – 4 = 11

Question 7.

Consider

i. Find A^{-1} and A^{T}

ii. Verify (A^{T})^{-1} = (A^{-1})^{T}

Answer.

Question 8.

Let

(i) Is A singular?

(ii) Find Adj A.

(iii) Find A^{-1}.

Answer.

(i)|A|= 1(4 – 0) – 4( – 2 – 0)

= 4 + 8 = 12

Since |A|≠0, A is non-singular.

Question 9.

Evaluate

Answer.

Question 10.

Let

Where 0≤θ≤2π. Then find Det(A)

Answer.

**Very Long Answer Type Questions**

**(Score 6)**

Question 1.

Consider the system of equations

x – y + z = 3

2x + y – z = 2

– x – 2y + 2z = 1

(i) Convert this system of equations in the standard form AX = B.

(ii) Is A invertible ?

(iii) Show that the system of equations is consistent and find the solution.

Answer.

Question 2.

Given

i. Express this as sum of two determinants.

ii. Prove that

∆ = (1+pxyz)(x-y)(y-z)(z-x)

Answer.

Expanding along C1,

= (1+pxyz) (y-x) (z-x) (z+x-y-x)

= (1+pxyz) (y-x) (z-x) (ziy)

= (1+pxyz) (x-y) (y-z) (z-x)

Question 3.

Ashok purchased 3 pencils, 2 instrument boxes and 1 pen and paid Rs 41. From the same shop, Babu purchased 2 pencils, 1 instrument box and 2 pens and paid Rs. 29 while Rajesh purchased 2 pencil, 2 instrument boxes and 2 pens and paid Rs. 44.

(i) Formulate the problem into a system of linear equations.

(ii) Find the cost of 1 pencil, 1 instrument box and 1 pen.

Answer.

(i) Let price of one pencil = Rs x

Let price of one box = Rs y

Let price of one pen = Rs z

Then 3x + 2y + z = 41

2x + y + 2z = 29

2x + 2y +2z = 44

(ii) In Matrix form, AX = B

Question 4.

If

(i) Is A non-singular? Why?

(ii) Find A^{-1}

(iii) Verify that (Adj A)^{-1} = (Adj A^{-1})

(iii) Solve the system of linear equations x + 2y – 3z = -4;

2x + 3y + 2z = 2; 3x – 3y – 4z = 11

Question 5.

Answer.

We know that, (AB)^{-1 }= B^{-1}A^{-1} and A^{-1} is known, therefore we proceed to find B^{-1}

Here,

**Edumate Questions & Answers**

Question 1.

i. Choose the correct answer from the brackets.If

then

a.0

b.-40

c.40

d.2

ii. Show that

Answer.

i. c.40

ii.

Question 2.

i. Choose the correct answer from the bracket. Let the value of a determinant is ∆,. Then the value of a determinant obtained by interchanging two rows is

{(a) ∆, (b)-∆, (c) 0, (d) 1}

ii. Show that

Answer.

i. (b) -∆

ii. Operator C_{1}→C_{1}+C_{2}+C_{3}, we have

Question 3.

i. The value of the determinant

ii. Using the properties of determinant, show that

Answer.

i. b . since , sin10cos50 + cos10sin50

= (a+b+c)(b-a) (c-a) (c+a-b-a)

= (a+b+c)(b-a) (c-a) (c-b)

= (b-c) (c-a) (a-b) (a+b+c)

Question 4.

i. Choose the correct answer from the bracket. The value of the determinant.

{(a) p+q+r, (b) 1, (c) 0, (d) 3pqr}

ii. Evaluate

Answer.

i.c.0 (since the given determinant is the determinant of a third order skew symmetric matrix)

Question 5.

i. Choose the correct answer from the bracket. Consider a square matrix of order 3.

Let C_{11}, C_{12}, C_{13} are co factors of the elements a_{11}, a_{12}, a_{13}, respectively then a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}, is

a. 0

b. |A|

c. 1

d. none of these

Answer.

i. b. |A|

ii.

Question 6.

i. Choose the correct answer from the bracket. If A = and A(adjA) = , then the value of k is …

{(a) 0, (b) 3, (c) 1, (d) 2)

ii. Find the Inverse of the matrix

Answer.

Question 7.

i. Choose the correct answer from the bracket. If A = and A^{-1} = kA, then the value of ‘k’ is

(a) find A²

(b) show that A² = A^{-1}

Answer.

Question 8.

i. Choose the correct answer from the bracket. If each element of a third order square matrix ‘A’ is multiplied by 3, then the determinant of the newly formed matrix is

{(a)9|A| (b)3|A|

(c) 27|A| (d) (|A|)^{3}}

ii. Consider the matrix A =

a. Show that ‘A’ satisfies the equation x² + 4x – 42 = 0.

b. Hence find A^{-1}.

Answer.

i. c) 27|A|

Question 9.

i. If A and B are matrices of order 3 such that |A| = -1,|B| = 3, then |3AB| is

(a) -9, (b) -27, (c) 41, (d) 9

ii.

Answer.

i. c) -81 (since|3AB| = 27|A||B|)

Question 10.

i. If what ¡s the value of |3A|?

ii. Find the equation of the line joining the points (1,2) and (-3,-2) using determinants.

Answer.

|3A| = 3³ |A|

= 27(- 2 + 3) = 27

ii. Let (x,y) be the coordinate of any point on the line, then (1,2), (-3,-2) and (x,y) are collinear.

Hence the area formed will be zero.

⇒ ( – 2 – y) – 2( – 3 – x) + 1 ( – 3y + 2x) = 0

⇒ 4x – 4y + 40 = 0

⇒ x – y + 1 = 0

Question 11.

i. Find the values of x in which

ii. Using the property of determinants, show that the points

A(a,b+c), B(b, c+a), C(c,a+b) are collinear.

iii. Examine the consistency of system of following equations.

5x4y+4z = 15, 7z+y-3z = 19, 2x+y+6z = 46

Answer.

Question 12.

If

i) Find |A|

ii) Find A^{-1}

iii) Solve the linear equations

3x-2y+3z = 8, 2x+y-z = 1, 4x-3y+2z = 4

Answer.

Question 13.

‘Arjun’ purchased 3 pens, 2 purses and 1 Instrument box and pays Rs. 410. From the same Shop ‘Deeraj’ purchases 2 pens, 1 purse and 2 instrument boxes and pays Rs. 290, while ‘Sindhu’ purchases 2 pens, 2 purses, 2 instrument boxes and pays Rs. 440.

i. Translate the equation into system of linear equations.

ii. The cost of pen, one purse and one Instrument box using matrix method.

Answer.

i. Let the price of one pen os Rs.x, one purse is Rs. y and one instrument box be Rs. z

3x+2y+z = 410 ; 2x+y+2z = 290; 2x = 2y+2z = 440

ii. The system can be represented by the matrix equation AX = B

Hence the cost one pen is Rs. 20, one purse is Rs. 150 and one instrument box is Rs. 50.

**NCERT Questions & Answers**

Question 1.

Write the value of x+y+z, if

Answer.

Question 2.

Using properties of determinants show that

Answer.

Applying R1 → R1 + R2 + R3, we get

Question 3.

Consider the points (b, c + a), (c, a+ b) and (a, b + c).

(i) Find the area of the triangle formed by these points.

(ii) Are the given points collinear? Why?

Answer

(ii) Since area of the triangle is zero, the points are collinear .

Question 4.

If

(i) Perform R1 → R1 + R2 + R3 on ∆

(ii) Take (a+b+c) common from R1 & rewrite ∆.

(iii) Show that ∆ = (a+b+c)³

Answer.

Question 5.

Given

(i) Express given determinant as the sum of 8 determinants

(ii) Show that

Answer.

Question 6.

Find equation of line joining (1,2) and (3,6) using determinants.

Answer.

Let (x,y ) be a point on the line, then the points ( x,y ), (1,2) and (3,6) are collinear,

⇒x(2-6) – y(1-3) + 1(6-6) = 0

⇒4x+2y = 0

⇒2x-y = 0 ⇒ y=2x

Question 7.

i. Evaluate ∆ by expanding along R1

ii.Prove that ∆ = x³

Answer.

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